退化抛物-双曲方程近似解的误差分析

退化抛物-双曲方程是一类非常著名的发展方程，在环境科学、能源开发、流体力学、电子科学和金融工程等领域具有广泛的应用。由于该方程在某些区域完全退化成双曲守恒律方程，解可能会出现激波，给理论研究和数值模拟增加了难度。最近，基于误差估计的自适应算法蓬勃发展，近似解的误差分析越来越重要。因此，本项目拟在双变量法和动力学框架的基础上，充分利用抛物的特性，深入研究退化抛物-双曲方程近似解与熵解的误差，建立两种Dirichlet问题粘性解和熵解的误差估计，为数值解的研究奠定理论基础；建立Cauchy问题的一些数值解和熵解的误差估计，为自适应算法的后验误差估计提供技术支持，推动退化抛物-双曲方程数值模拟的发展。

Degenerate parabolic-hyperbolic equations，which are a kind of very famous evolution equations, have been extensively used in the fields of environmental science, energy development, fluid mechanics,electronic science, finance engineering and so on. Since it can be scalar conservation law on some region, its solution may be discontinuous, thus theoretical research and numerical simulation are very difficult. Recently, adaptive algorithms on the basis of error estimate are booming, and error analysis of approximate solutions is more important. Therefore, this project is aimed to combine the parabolic features effectively to creatively improve doubling variable device and kinetic formulation to deeply study error between its approximate solutions and entropy solutions. The main objective is to prove error estimate of viscosity solutions and entropy solutions for two Dirichlet problems in order to establish theoretical foundation of numerical analysis, and to establish error estimate of some numerical solutions and entropy solution for Cauchy problems in order to give technical support for posteriori error estimates of adaptive algorithms.